\(\underline{\mbox{Subcase 2.2.}}\) \(g=4\) .
Since the least eigenvalue of the principle submatrix indexed by \(V(H_2)\)  is always less than \(-2\) , we can get \(\Sigma \)  is a signed complete bipartite graph through the same methods in [1]}. And then \(\Sigma \)  is balanced by Claim REF  of Subcase 2.1 immediately.
